Problem: All of the 5th grade teachers and students from Santa Rita went on a field trip to an archaeology museum. Tickets were $$5.00$ each for teachers and $$4.50$ each for students, and the group paid $$42.00$ in total. The next month, the same group visited a science museum where the tickets cost $$15.00$ each for teachers and $$9.50$ each for students, and the group paid $$102.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+4.5y = 42}$ ${15x+9.5y = 102}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-15x-13.5y = -126}$ ${15x+9.5y = 102}$ Add the top and bottom equations together. $ -4y = -24 $ $ y = \dfrac{-24}{-4}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {5x+4.5y = 42}$ to find $x$ ${5x + 4.5}{(6)}{= 42}$ $5x+27 = 42$ $5x = 15$ $x = \dfrac{15}{5}$ ${x = 3}$ You can also plug ${y = 6}$ into $ {15x+9.5y = 102}$ and get the same answer for $x$ ${15x + 9.5}{(6)}{= 102}$ ${x = 3}$ There were $3$ teachers and $6$ students on the field trips.